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Theorem ltexprlemloc 7597
Description: Our constructed difference is located. Lemma for ltexpri 7603. (Contributed by Jim Kingdon, 17-Dec-2019.)
Hypothesis
Ref Expression
ltexprlem.1  |-  C  = 
<. { x  e.  Q.  |  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  {
x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.
Assertion
Ref Expression
ltexprlemloc  |-  ( A 
<P  B  ->  A. q  e.  Q.  A. r  e. 
Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  C
)  \/  r  e.  ( 2nd `  C
) ) ) )
Distinct variable groups:    x, y, q, r, A    x, B, y, q, r    x, C, y, q, r

Proof of Theorem ltexprlemloc
Dummy variables  z  w  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltexnqi 7399 . . . . . 6  |-  ( q 
<Q  r  ->  E. w  e.  Q.  ( q  +Q  w )  =  r )
21adantl 277 . . . . 5  |-  ( ( A  <P  B  /\  q  <Q  r )  ->  E. w  e.  Q.  ( q  +Q  w
)  =  r )
3 ltrelpr 7495 . . . . . . . . . 10  |-  <P  C_  ( P.  X.  P. )
43brel 4675 . . . . . . . . 9  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
54simpld 112 . . . . . . . 8  |-  ( A 
<P  B  ->  A  e. 
P. )
6 prop 7465 . . . . . . . . 9  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
7 prarloc 7493 . . . . . . . . 9  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  w  e.  Q. )  ->  E. z  e.  ( 1st `  A ) E. y  e.  ( 2nd `  A ) y  <Q  ( z  +Q  w ) )
86, 7sylan 283 . . . . . . . 8  |-  ( ( A  e.  P.  /\  w  e.  Q. )  ->  E. z  e.  ( 1st `  A ) E. y  e.  ( 2nd `  A ) y  <Q  ( z  +Q  w ) )
95, 8sylan 283 . . . . . . 7  |-  ( ( A  <P  B  /\  w  e.  Q. )  ->  E. z  e.  ( 1st `  A ) E. y  e.  ( 2nd `  A ) y  <Q  ( z  +Q  w ) )
109ad2ant2r 509 . . . . . 6  |-  ( ( ( A  <P  B  /\  q  <Q  r )  /\  ( w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  ->  E. z  e.  ( 1st `  A
) E. y  e.  ( 2nd `  A
) y  <Q  (
z  +Q  w ) )
114simprd 114 . . . . . . . . . . . . . 14  |-  ( A 
<P  B  ->  B  e. 
P. )
1211ad2antrr 488 . . . . . . . . . . . . 13  |-  ( ( ( A  <P  B  /\  q  <Q  r )  /\  ( w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  ->  B  e.  P. )
1312ad2antrr 488 . . . . . . . . . . . 12  |-  ( ( ( ( ( A 
<P  B  /\  q  <Q  r )  /\  (
w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  /\  ( z  e.  ( 1st `  A
)  /\  y  e.  ( 2nd `  A ) ) )  /\  y  <Q  ( z  +Q  w
) )  ->  B  e.  P. )
14 ltanqg 7390 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
1514adantl 277 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A 
<P  B  /\  q  <Q  r )  /\  (
w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  /\  ( z  e.  ( 1st `  A
)  /\  y  e.  ( 2nd `  A ) ) )  /\  (
f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. ) )  -> 
( f  <Q  g  <->  ( h  +Q  f ) 
<Q  ( h  +Q  g
) ) )
16 elprnqu 7472 . . . . . . . . . . . . . . . . . . 19  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
176, 16sylan 283 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
185, 17sylan 283 . . . . . . . . . . . . . . . . 17  |-  ( ( A  <P  B  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
1918adantlr 477 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  <P  B  /\  q  <Q  r )  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
2019ad2ant2rl 511 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  y  e.  Q. )
21 elprnql 7471 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
226, 21sylan 283 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
235, 22sylan 283 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  <P  B  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
2423adantlr 477 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  <P  B  /\  q  <Q  r )  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
2524ad2ant2r 509 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  z  e.  Q. )
26 simplrl 535 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  w  e.  Q. )
27 addclnq 7365 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  Q.  /\  w  e.  Q. )  ->  ( z  +Q  w
)  e.  Q. )
2825, 26, 27syl2anc 411 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  (
z  +Q  w )  e.  Q. )
29 ltrelnq 7355 . . . . . . . . . . . . . . . . . . 19  |-  <Q  C_  ( Q.  X.  Q. )
3029brel 4675 . . . . . . . . . . . . . . . . . 18  |-  ( q 
<Q  r  ->  ( q  e.  Q.  /\  r  e.  Q. ) )
3130simpld 112 . . . . . . . . . . . . . . . . 17  |-  ( q 
<Q  r  ->  q  e. 
Q. )
3231adantl 277 . . . . . . . . . . . . . . . 16  |-  ( ( A  <P  B  /\  q  <Q  r )  -> 
q  e.  Q. )
3332ad2antrr 488 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  q  e.  Q. )
34 addcomnqg 7371 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
3534adantl 277 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A 
<P  B  /\  q  <Q  r )  /\  (
w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  /\  ( z  e.  ( 1st `  A
)  /\  y  e.  ( 2nd `  A ) ) )  /\  (
f  e.  Q.  /\  g  e.  Q. )
)  ->  ( f  +Q  g )  =  ( g  +Q  f ) )
3615, 20, 28, 33, 35caovord2d 6038 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  (
y  <Q  ( z  +Q  w )  <->  ( y  +Q  q )  <Q  (
( z  +Q  w
)  +Q  q ) ) )
37 addassnqg 7372 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  Q.  /\  w  e.  Q.  /\  q  e.  Q. )  ->  (
( z  +Q  w
)  +Q  q )  =  ( z  +Q  ( w  +Q  q
) ) )
3825, 26, 33, 37syl3anc 1238 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  (
( z  +Q  w
)  +Q  q )  =  ( z  +Q  ( w  +Q  q
) ) )
39 addcomnqg 7371 . . . . . . . . . . . . . . . . . 18  |-  ( ( w  e.  Q.  /\  q  e.  Q. )  ->  ( w  +Q  q
)  =  ( q  +Q  w ) )
4026, 33, 39syl2anc 411 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  (
w  +Q  q )  =  ( q  +Q  w ) )
4140oveq2d 5885 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  (
z  +Q  ( w  +Q  q ) )  =  ( z  +Q  ( q  +Q  w
) ) )
42 simplrr 536 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  (
q  +Q  w )  =  r )
4342oveq2d 5885 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  (
z  +Q  ( q  +Q  w ) )  =  ( z  +Q  r ) )
4438, 41, 433eqtrd 2214 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  (
( z  +Q  w
)  +Q  q )  =  ( z  +Q  r ) )
4544breq2d 4012 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  (
( y  +Q  q
)  <Q  ( ( z  +Q  w )  +Q  q )  <->  ( y  +Q  q )  <Q  (
z  +Q  r ) ) )
4636, 45bitrd 188 . . . . . . . . . . . . 13  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  (
y  <Q  ( z  +Q  w )  <->  ( y  +Q  q )  <Q  (
z  +Q  r ) ) )
4746biimpa 296 . . . . . . . . . . . 12  |-  ( ( ( ( ( A 
<P  B  /\  q  <Q  r )  /\  (
w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  /\  ( z  e.  ( 1st `  A
)  /\  y  e.  ( 2nd `  A ) ) )  /\  y  <Q  ( z  +Q  w
) )  ->  (
y  +Q  q ) 
<Q  ( z  +Q  r
) )
48 prop 7465 . . . . . . . . . . . . 13  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
49 prloc 7481 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  ( y  +Q  q
)  <Q  ( z  +Q  r ) )  -> 
( ( y  +Q  q )  e.  ( 1st `  B )  \/  ( z  +Q  r )  e.  ( 2nd `  B ) ) )
5048, 49sylan 283 . . . . . . . . . . . 12  |-  ( ( B  e.  P.  /\  ( y  +Q  q
)  <Q  ( z  +Q  r ) )  -> 
( ( y  +Q  q )  e.  ( 1st `  B )  \/  ( z  +Q  r )  e.  ( 2nd `  B ) ) )
5113, 47, 50syl2anc 411 . . . . . . . . . . 11  |-  ( ( ( ( ( A 
<P  B  /\  q  <Q  r )  /\  (
w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  /\  ( z  e.  ( 1st `  A
)  /\  y  e.  ( 2nd `  A ) ) )  /\  y  <Q  ( z  +Q  w
) )  ->  (
( y  +Q  q
)  e.  ( 1st `  B )  \/  (
z  +Q  r )  e.  ( 2nd `  B
) ) )
5251ex 115 . . . . . . . . . 10  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  (
y  <Q  ( z  +Q  w )  ->  (
( y  +Q  q
)  e.  ( 1st `  B )  \/  (
z  +Q  r )  e.  ( 2nd `  B
) ) ) )
5352anassrs 400 . . . . . . . . 9  |-  ( ( ( ( ( A 
<P  B  /\  q  <Q  r )  /\  (
w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  /\  z  e.  ( 1st `  A
) )  /\  y  e.  ( 2nd `  A
) )  ->  (
y  <Q  ( z  +Q  w )  ->  (
( y  +Q  q
)  e.  ( 1st `  B )  \/  (
z  +Q  r )  e.  ( 2nd `  B
) ) ) )
5453reximdva 2579 . . . . . . . 8  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  z  e.  ( 1st `  A ) )  -> 
( E. y  e.  ( 2nd `  A
) y  <Q  (
z  +Q  w )  ->  E. y  e.  ( 2nd `  A ) ( ( y  +Q  q )  e.  ( 1st `  B )  \/  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
5554reximdva 2579 . . . . . . 7  |-  ( ( ( A  <P  B  /\  q  <Q  r )  /\  ( w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  ->  ( E. z  e.  ( 1st `  A ) E. y  e.  ( 2nd `  A
) y  <Q  (
z  +Q  w )  ->  E. z  e.  ( 1st `  A ) E. y  e.  ( 2nd `  A ) ( ( y  +Q  q )  e.  ( 1st `  B )  \/  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
56 prml 7467 . . . . . . . . . . . 12  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  ->  E. z  e.  Q.  z  e.  ( 1st `  A ) )
57 rexex 2523 . . . . . . . . . . . 12  |-  ( E. z  e.  Q.  z  e.  ( 1st `  A
)  ->  E. z 
z  e.  ( 1st `  A ) )
586, 56, 573syl 17 . . . . . . . . . . 11  |-  ( A  e.  P.  ->  E. z 
z  e.  ( 1st `  A ) )
59 r19.45mv 3516 . . . . . . . . . . 11  |-  ( E. z  z  e.  ( 1st `  A )  ->  ( E. z  e.  ( 1st `  A
) ( E. y  e.  ( 2nd `  A
) ( y  +Q  q )  e.  ( 1st `  B )  \/  ( z  +Q  r )  e.  ( 2nd `  B ) )  <->  ( E. y  e.  ( 2nd `  A
) ( y  +Q  q )  e.  ( 1st `  B )  \/  E. z  e.  ( 1st `  A
) ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
605, 58, 593syl 17 . . . . . . . . . 10  |-  ( A 
<P  B  ->  ( E. z  e.  ( 1st `  A ) ( E. y  e.  ( 2nd `  A ) ( y  +Q  q )  e.  ( 1st `  B
)  \/  ( z  +Q  r )  e.  ( 2nd `  B
) )  <->  ( E. y  e.  ( 2nd `  A ) ( y  +Q  q )  e.  ( 1st `  B
)  \/  E. z  e.  ( 1st `  A
) ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
6160adantr 276 . . . . . . . . 9  |-  ( ( A  <P  B  /\  q  <Q  r )  -> 
( E. z  e.  ( 1st `  A
) ( E. y  e.  ( 2nd `  A
) ( y  +Q  q )  e.  ( 1st `  B )  \/  ( z  +Q  r )  e.  ( 2nd `  B ) )  <->  ( E. y  e.  ( 2nd `  A
) ( y  +Q  q )  e.  ( 1st `  B )  \/  E. z  e.  ( 1st `  A
) ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
62 prmu 7468 . . . . . . . . . . . . 13  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  ->  E. x  e.  Q.  x  e.  ( 2nd `  A ) )
63 rexex 2523 . . . . . . . . . . . . 13  |-  ( E. x  e.  Q.  x  e.  ( 2nd `  A
)  ->  E. x  x  e.  ( 2nd `  A ) )
646, 62, 633syl 17 . . . . . . . . . . . 12  |-  ( A  e.  P.  ->  E. x  x  e.  ( 2nd `  A ) )
65 r19.43 2635 . . . . . . . . . . . . 13  |-  ( E. y  e.  ( 2nd `  A ) ( ( y  +Q  q )  e.  ( 1st `  B
)  \/  ( z  +Q  r )  e.  ( 2nd `  B
) )  <->  ( E. y  e.  ( 2nd `  A ) ( y  +Q  q )  e.  ( 1st `  B
)  \/  E. y  e.  ( 2nd `  A
) ( z  +Q  r )  e.  ( 2nd `  B ) ) )
66 r19.9rmv 3514 . . . . . . . . . . . . . 14  |-  ( E. x  x  e.  ( 2nd `  A )  ->  ( ( z  +Q  r )  e.  ( 2nd `  B
)  <->  E. y  e.  ( 2nd `  A ) ( z  +Q  r
)  e.  ( 2nd `  B ) ) )
6766orbi2d 790 . . . . . . . . . . . . 13  |-  ( E. x  x  e.  ( 2nd `  A )  ->  ( ( E. y  e.  ( 2nd `  A ) ( y  +Q  q )  e.  ( 1st `  B
)  \/  ( z  +Q  r )  e.  ( 2nd `  B
) )  <->  ( E. y  e.  ( 2nd `  A ) ( y  +Q  q )  e.  ( 1st `  B
)  \/  E. y  e.  ( 2nd `  A
) ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
6865, 67bitr4id 199 . . . . . . . . . . . 12  |-  ( E. x  x  e.  ( 2nd `  A )  ->  ( E. y  e.  ( 2nd `  A
) ( ( y  +Q  q )  e.  ( 1st `  B
)  \/  ( z  +Q  r )  e.  ( 2nd `  B
) )  <->  ( E. y  e.  ( 2nd `  A ) ( y  +Q  q )  e.  ( 1st `  B
)  \/  ( z  +Q  r )  e.  ( 2nd `  B
) ) ) )
695, 64, 683syl 17 . . . . . . . . . . 11  |-  ( A 
<P  B  ->  ( E. y  e.  ( 2nd `  A ) ( ( y  +Q  q )  e.  ( 1st `  B
)  \/  ( z  +Q  r )  e.  ( 2nd `  B
) )  <->  ( E. y  e.  ( 2nd `  A ) ( y  +Q  q )  e.  ( 1st `  B
)  \/  ( z  +Q  r )  e.  ( 2nd `  B
) ) ) )
7069rexbidv 2478 . . . . . . . . . 10  |-  ( A 
<P  B  ->  ( E. z  e.  ( 1st `  A ) E. y  e.  ( 2nd `  A
) ( ( y  +Q  q )  e.  ( 1st `  B
)  \/  ( z  +Q  r )  e.  ( 2nd `  B
) )  <->  E. z  e.  ( 1st `  A
) ( E. y  e.  ( 2nd `  A
) ( y  +Q  q )  e.  ( 1st `  B )  \/  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
7170adantr 276 . . . . . . . . 9  |-  ( ( A  <P  B  /\  q  <Q  r )  -> 
( E. z  e.  ( 1st `  A
) E. y  e.  ( 2nd `  A
) ( ( y  +Q  q )  e.  ( 1st `  B
)  \/  ( z  +Q  r )  e.  ( 2nd `  B
) )  <->  E. z  e.  ( 1st `  A
) ( E. y  e.  ( 2nd `  A
) ( y  +Q  q )  e.  ( 1st `  B )  \/  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
72 ltexprlem.1 . . . . . . . . . . . . . 14  |-  C  = 
<. { x  e.  Q.  |  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  {
x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.
7372ltexprlemell 7588 . . . . . . . . . . . . 13  |-  ( q  e.  ( 1st `  C
)  <->  ( q  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )
7472ltexprlemelu 7589 . . . . . . . . . . . . . 14  |-  ( r  e.  ( 2nd `  C
)  <->  ( r  e. 
Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )
75 eleq1 2240 . . . . . . . . . . . . . . . . 17  |-  ( y  =  z  ->  (
y  e.  ( 1st `  A )  <->  z  e.  ( 1st `  A ) ) )
76 oveq1 5876 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  z  ->  (
y  +Q  r )  =  ( z  +Q  r ) )
7776eleq1d 2246 . . . . . . . . . . . . . . . . 17  |-  ( y  =  z  ->  (
( y  +Q  r
)  e.  ( 2nd `  B )  <->  ( z  +Q  r )  e.  ( 2nd `  B ) ) )
7875, 77anbi12d 473 . . . . . . . . . . . . . . . 16  |-  ( y  =  z  ->  (
( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) )  <->  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
7978cbvexv 1918 . . . . . . . . . . . . . . 15  |-  ( E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) )  <->  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) )
8079anbi2i 457 . . . . . . . . . . . . . 14  |-  ( ( r  e.  Q.  /\  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) )  <->  ( r  e.  Q.  /\  E. z
( z  e.  ( 1st `  A )  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
8174, 80bitri 184 . . . . . . . . . . . . 13  |-  ( r  e.  ( 2nd `  C
)  <->  ( r  e. 
Q.  /\  E. z
( z  e.  ( 1st `  A )  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
8273, 81orbi12i 764 . . . . . . . . . . . 12  |-  ( ( q  e.  ( 1st `  C )  \/  r  e.  ( 2nd `  C
) )  <->  ( (
q  e.  Q.  /\  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) )  \/  (
r  e.  Q.  /\  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) ) )
83 ibar 301 . . . . . . . . . . . . . . 15  |-  ( q  e.  Q.  ->  ( E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  <->  ( q  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) ) )
8483adantr 276 . . . . . . . . . . . . . 14  |-  ( ( q  e.  Q.  /\  r  e.  Q. )  ->  ( E. y ( y  e.  ( 2nd `  A )  /\  (
y  +Q  q )  e.  ( 1st `  B
) )  <->  ( q  e.  Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) ) )
85 ibar 301 . . . . . . . . . . . . . . 15  |-  ( r  e.  Q.  ->  ( E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  r )  e.  ( 2nd `  B ) )  <->  ( r  e. 
Q.  /\  E. z
( z  e.  ( 1st `  A )  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) ) )
8685adantl 277 . . . . . . . . . . . . . 14  |-  ( ( q  e.  Q.  /\  r  e.  Q. )  ->  ( E. z ( z  e.  ( 1st `  A )  /\  (
z  +Q  r )  e.  ( 2nd `  B
) )  <->  ( r  e.  Q.  /\  E. z
( z  e.  ( 1st `  A )  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) ) )
8784, 86orbi12d 793 . . . . . . . . . . . . 13  |-  ( ( q  e.  Q.  /\  r  e.  Q. )  ->  ( ( E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  \/  E. z
( z  e.  ( 1st `  A )  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) )  <->  ( (
q  e.  Q.  /\  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) )  \/  (
r  e.  Q.  /\  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) ) ) )
8830, 87syl 14 . . . . . . . . . . . 12  |-  ( q 
<Q  r  ->  ( ( E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  \/  E. z
( z  e.  ( 1st `  A )  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) )  <->  ( (
q  e.  Q.  /\  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) )  \/  (
r  e.  Q.  /\  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) ) ) )
8982, 88bitr4id 199 . . . . . . . . . . 11  |-  ( q 
<Q  r  ->  ( ( q  e.  ( 1st `  C )  \/  r  e.  ( 2nd `  C
) )  <->  ( E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  \/  E. z
( z  e.  ( 1st `  A )  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) ) )
90 df-rex 2461 . . . . . . . . . . . 12  |-  ( E. y  e.  ( 2nd `  A ) ( y  +Q  q )  e.  ( 1st `  B
)  <->  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) )
91 df-rex 2461 . . . . . . . . . . . 12  |-  ( E. z  e.  ( 1st `  A ) ( z  +Q  r )  e.  ( 2nd `  B
)  <->  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) )
9290, 91orbi12i 764 . . . . . . . . . . 11  |-  ( ( E. y  e.  ( 2nd `  A ) ( y  +Q  q
)  e.  ( 1st `  B )  \/  E. z  e.  ( 1st `  A ) ( z  +Q  r )  e.  ( 2nd `  B
) )  <->  ( E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  \/  E. z
( z  e.  ( 1st `  A )  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
9389, 92bitr4di 198 . . . . . . . . . 10  |-  ( q 
<Q  r  ->  ( ( q  e.  ( 1st `  C )  \/  r  e.  ( 2nd `  C
) )  <->  ( E. y  e.  ( 2nd `  A ) ( y  +Q  q )  e.  ( 1st `  B
)  \/  E. z  e.  ( 1st `  A
) ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
9493adantl 277 . . . . . . . . 9  |-  ( ( A  <P  B  /\  q  <Q  r )  -> 
( ( q  e.  ( 1st `  C
)  \/  r  e.  ( 2nd `  C
) )  <->  ( E. y  e.  ( 2nd `  A ) ( y  +Q  q )  e.  ( 1st `  B
)  \/  E. z  e.  ( 1st `  A
) ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
9561, 71, 943bitr4rd 221 . . . . . . . 8  |-  ( ( A  <P  B  /\  q  <Q  r )  -> 
( ( q  e.  ( 1st `  C
)  \/  r  e.  ( 2nd `  C
) )  <->  E. z  e.  ( 1st `  A
) E. y  e.  ( 2nd `  A
) ( ( y  +Q  q )  e.  ( 1st `  B
)  \/  ( z  +Q  r )  e.  ( 2nd `  B
) ) ) )
9695adantr 276 . . . . . . 7  |-  ( ( ( A  <P  B  /\  q  <Q  r )  /\  ( w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  ->  ( (
q  e.  ( 1st `  C )  \/  r  e.  ( 2nd `  C
) )  <->  E. z  e.  ( 1st `  A
) E. y  e.  ( 2nd `  A
) ( ( y  +Q  q )  e.  ( 1st `  B
)  \/  ( z  +Q  r )  e.  ( 2nd `  B
) ) ) )
9755, 96sylibrd 169 . . . . . 6  |-  ( ( ( A  <P  B  /\  q  <Q  r )  /\  ( w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  ->  ( E. z  e.  ( 1st `  A ) E. y  e.  ( 2nd `  A
) y  <Q  (
z  +Q  w )  ->  ( q  e.  ( 1st `  C
)  \/  r  e.  ( 2nd `  C
) ) ) )
9810, 97mpd 13 . . . . 5  |-  ( ( ( A  <P  B  /\  q  <Q  r )  /\  ( w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  ->  ( q  e.  ( 1st `  C
)  \/  r  e.  ( 2nd `  C
) ) )
992, 98rexlimddv 2599 . . . 4  |-  ( ( A  <P  B  /\  q  <Q  r )  -> 
( q  e.  ( 1st `  C )  \/  r  e.  ( 2nd `  C ) ) )
10099ex 115 . . 3  |-  ( A 
<P  B  ->  ( q 
<Q  r  ->  ( q  e.  ( 1st `  C
)  \/  r  e.  ( 2nd `  C
) ) ) )
101100ralrimivw 2551 . 2  |-  ( A 
<P  B  ->  A. r  e.  Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  C
)  \/  r  e.  ( 2nd `  C
) ) ) )
102101ralrimivw 2551 1  |-  ( A 
<P  B  ->  A. q  e.  Q.  A. r  e. 
Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  C
)  \/  r  e.  ( 2nd `  C
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708    /\ w3a 978    = wceq 1353   E.wex 1492    e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459   <.cop 3594   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   1stc1st 6133   2ndc2nd 6134   Q.cnq 7270    +Q cplq 7272    <Q cltq 7275   P.cnp 7281    <P cltp 7285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-iltp 7460
This theorem is referenced by:  ltexprlempr  7598
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